Contents

Idea

Two morphismsC→LDC \stackrel{L}{\to} D and D→RCD \stackrel{R}{\to} C in a 2-category𝒞\mathcal{C} form an adjunction if they are dual to each other.

There are two archetypical examples:

If AA is a monoidal category and 𝒞=BA\mathcal{C} = \mathbf{B}A is the one-object 2-category incarnation of AA (the delooping of AA), so that the morphisms in 𝒞\mathcal{C} correspond to the objects of AA, then the notion of adjoint morphisms in 𝒞\mathcal{C} coincides precisely with the notion of dual objects in a AA.

If 𝒞\mathcal{C} is the 22-category Cat, so that the morphisms in 𝒞\mathcal{C} are functors, then the notion of adjoint morphisms in 𝒞\mathcal{C} coincides precisely with the notion of adjoint functors.

General

The notion of adjunction may usefully be thought of as a weakened version of the notion of equivalence in a 2-category: a morphism in an adjunction need not be invertible, but it has in some sense a left inverse from below and a right inverse from above. If the morphism in an adjunction does happen to be a genuine equivalence, then we speak of the adjunction being an adjoint equivalence.

Essentially everything that makes category theory nontrivial and interesting beyond groupoid theory can be derived from the concept of adjoint functors. In particular universal constructions such as limits and colimits are examples of certain adjunctions. Adjunctions are already interesting (but simpler) in 2-posets, such as the 22-poset Pos of posets.

From hom-functors to units and counits

At the cost of some repetition (compare adjoint functor), we outline how one gets from the hom-functor formulation of adjunction in Cat to the elementary definition in terms of units and counits. This will motivate the definition in the section that follows, which is elementary (definable in the first-order theory of categories) and portable to any 2-category.

We start from a familiar example. Let U:Grp→SetU: Grp \to Set from groups to sets be the usual forgetful functor. When we say “F(X)F(X) is the free group generated by a set XX”, we mean there is a function ηX:X→U(F(X))\eta_X: X \to U(F(X)) which is universal among functions from XX to the underlying set of a group, which means in turn that given a function f:X→U(G)f: X \to U(G), there is a unique group homomorphism g:F(X)→Gg: F(X) \to G such that

Here ηX\eta_X is a component of what we call the unit of the adjunctionF⊣UF \dashv U, and the equation above is a recipe for the relationship between the map g:F(X)→Gg: F(X) \to G and the map f:X→U(G)f: X \to U(G) in terms of the unit.

Now we work more generally. Suppose given functors L:C→DL: C \to D, R:D→CR: D \to C and the structure of an adjunction in the form of a natural isomorphism

Now the idea is that, a la the Yoneda lemma, Ψ\Psi should be completely describable in terms of what it does to identity maps. With that in mind, define the unit η:1C→RL\eta : 1_C \to R L by the formula ηc=Ψc,L(c)(1L(c))\eta_c = \Psi_{c, L(c)}(1_{L(c)}). Dually, define the counit ε:LR→1D\varepsilon : L R \to 1_D by the formula εd=ΨR(d),d−1(1R(d))\varepsilon_d = \Psi^{-1}_{R(d), d}(1_{R(d)}). Then given g:L(c)→dg: L(c) \to d, the claim is that

(In fact, we spell out the Yoneda-lemma proof of this dual form below.)

Finally, these operations should obviously be mutually inverse, but that can again be entirely encapsulated Yoneda-wise in terms of the effect on identity maps. Thus, if ηc≔Ψc,L(c)(1L(c))\eta_c \coloneqq \Psi_{c, L(c)}(1_{L(c)}), the recipe just given for Ψ−1\Psi^{-1} yields back

and this is one of the famous triangular equations: 1L=(L→LηLRL→εLL)1_L = (L \stackrel{L \eta}{\to} L R L \stackrel{\varepsilon L}{\to} L). Note that juxtaposition in the diagram above is neither functor application, nor vertical composition, nor horizontal composition, but is actually whiskering. By duality, we have the other triangular equation 1R=(R→ηRRLR→RεR)1_R = (R \stackrel{\eta R}{\to} R L R \stackrel{R \varepsilon}{\to} R). These two triangular equations are enough to guarantee that the recipes for Ψ\Psi and Ψ−1\Psi^{-1} are indeed mutually inverse.

Thus, it is perfectly sufficient to define an adjoint pair of functors in CatCat as given by unit and counit transformations η:1C→RL\eta: 1_C \to R L, ε:LR→1D\varepsilon: L R \to 1_D, satisfying triangular equations as above.

One thing often heard is that the definition of adjunctions via units and counits is an “elementary” definition (so that by implication, the formulation in terms of hom-functors is not elementary). This means that whereas the hom-functor formulation relies on a background category of sets, the formulation in terms of units and counits is purely in the first-order language of categories and makes no reference to a background model of set theory. It is therefore a perfectly serviceable definition of adjunction without assumptions of local smallness.

Yoneda-lemma argument

We claim that Ψc,d−1:homC(c,R(d))→homD(L(c),d)\Psi^{-1}_{c, d}: \hom_C(c, R(d)) \to \hom_D(L(c), d) can be defined by the formula

Chasing the element 1R(d)1_{R(d)} down and then across, we get f:c→R(d)f: c \to R(d) and then Ψc,d−1(f)\Psi^{-1}_{c, d}(f). Chasing across and then down, we get εd\varepsilon_d and then εd∘L(f)\varepsilon_d \circ L(f). This completes the verification of the claim.

When interpreted in the prototypical 2-category Cat, CC and DD are categories, LL and RR are functors, and η\eta and ϵ\epsilon are natural transformations. In this case (which was of course the first to be defined) there are a number of equivalent definitions of an adjunction, which can be found on the page adjoint functor. Conversely, the definition in any 2-category can be obtained by internalization from the definition in Cat\Cat.